Best-first search can be regarded as anytime scheme for producing lower bounds on the optimal solution, a characteristic that is mostly overlooked. We explore this topic in the context of AND/OR best-first search, guided by the MBE heuristic, when solving graphical models. In that context, the impact of the secondary heuristic for subproblem ordering may be significant, especially in the anytime context. Indeed, our paper illustrates this, showing that a new concept of bucket errors can advise in providing effective subproblem orderings in AND/OR search for both exact and anytime solutions.

We introduce the concept of local bucket error for the mini-bucket heuristics and show how it can be used to improve the power of AND/OR search for combinatorial optimization tasks in graphical models (e.g. MAP/MPE or weighted CSPs). The local bucket error illuminates how the heuristic errors are distributed in the search space, guided by the mini-bucket heuristic. We present and analyze methods for compiling the local bucket-errors (exactly and approximately) and show that they can be used to yield an effective tool for balancing look-ahead overhead during search. This can be especially instrumental when memory is restricted, accommodating the generation of only weak compiled heuristics. We illustrate the impact of the proposed schemes in an extensive empirical evaluation for both finding exact solutions and anytime suboptimal solutions.

Best-first search can be regarded as anytime scheme for producing lower bounds on the optimal solution, a characteristic that is mostly overlooked. We explore this topic in the context of AND/OR best-first search, guided by the MBE heuristic, when solving graphical models. In that context, the impact of the secondary heuristic for subproblem ordering may be significant, especially in the anytime context. Indeed, our paper illustrates this, showing that the new concept of bucket errors can advise in providing effective subproblem orderings in AND/OR search.

Many combinatorial problems are solved with a Depth-First search (DFS) guided by a heuristic and it is well-known that this method is very fragile with respect to heuristic mistakes. One standard way to make DFS more robust is to search by increasing number of discrepancies. This approach has been found useful in several domains where the search structure is a height-bounded OR tree. In this paper we investigate the generalization of discrepancy-based search to AND/OR search trees and propose an extension of the Limited Discrepancy Search (LDS) algorithm. We demonstrate the relevance of our proposal in the context of Graphical Models. In these problems, which can be solved with either a standard OR search tree or an AND/OR tree, we show the superiority of our approach. For a fixed number of discrepancies, the search space visited by the AND/OR algorithm strictly contains the search space visited by standard LDS, and many more nodes can be visited due to the multiplicative effect of the AND/OR decomposition. Besides, if the AND/OR tree achieves a significant size reduction with respect to the standard OR tree, the cost of each iteration of the AND/OR algorithm is asymptotically lower than in standard LDS. We report experiments on the minsum problem on different domains and show that the AND/OR version of LDS usually obtains better solutions given the same CPU time.

The paper investigates the potential of look-ahead in the con-text of AND/OR search in graphical models using the Mini-Bucket heuristic for combinatorial optimization tasks (e.g., MAP/MPE or weighted CSPs). We present and analyze the complexity of computing the residual (a.k.a Bellman update) of the Mini-Bucket heuristic and show how this can be used to identify which parts of the search space are more likely to benefit from look-ahead and how to bound its overhead. We also rephrase the look-ahead computation as a graphical model, to facilitate structure exploiting inference schemes. We demonstrate empirically that augmenting Mini-Bucket heuristics by look-ahead is a cost-effective way of increasing the power of Branch-And-Bound search.

Bounded Max-Sum is a message-passing algorithm for solving distributed Constraint Optimization Problems (DCOP) able to compute solutions with a guaranteed approximation ratio. In this paper we show that the introduction of an intermediate step that decomposes functions may significantly improve its accuracy. This is especially relevant in critical applications (e.g. automatic surveillance, disaster response scenarios) where the accuracy of solutions is of vital importance.

Graphical models are one of the most prominent frameworks to model complex systems and efficiently query them. Their underlying algebraic
properties are captured by a valuation structure that, most usually, is a
semiring. Depending on the semiring of choice, we can capture probabilistic models, constraint networks, cost networks, etc. In this paper we address the partitioning problem which occurs in many approximation techniques such as mini-bucket elimination and join-
graph propagation algorithms. Roghly speaking, subject to complexity bounds, the algorithm needs to find a partition of a set of factors such that best approximates the whole set. While this problem has been addressed in the past in a particular case, we present here a general description. Furthermore, we also propose a general partitioning scheme. Our proposal is general in the sense that it is presented
in terms of a generic semiring with the only additional requirements of a division operation and a refinement of its order. The proposed algorithm instantiates to the particular task of computing the
probability of evidence, but also applies directly to other important reasoning tasks. We demonstrate its good empirical behaviour on the problem of computing the most probable explanation.

Graphical models are one of the most prominent frameworks to model complex systems and efficiently query them. Their underlying algebraic properties are captured by a valuation structure that, most usually, is a semiring. Depending on the semiring of choice, we can capture probabilistic models, constraint networks, cost networks, etc. In this paper we address the partitioning problem which occurs in many approximation techniques such as mini-bucket elimination and join-graph propagation algorithms. Roghly speaking, subject to complexity bounds, the algorithm needs to find a partition of a set of factors such that best approximates the whole set. While this problem has been addressed in the past in a particular case, we present here a general description. Furthermore, we also propose a general partitioning scheme. Our proposal is general in the sense that it is presented
in terms of a generic semiring with the only additional requirements of a division operation and a refinement of its order. The proposed algorithm instantiates to the particular task of computing the
probability of evidence, but also applies directly to other important reasoning tasks. We demonstrate its good empirical behaviour on the problem of computing the most probable explanation.

Bounded Max-Sum is a message-passing algorithm for solving Distributes Constraint Optimization Problems (DCOP) able to compute solutions with a guaranteed approximation ratio. In this paper we show that the introduction of an intermediate step that decomposes functions may significantly improve its accuracy. This is especially relevant in critical applications (e.g. automatic surveillance, disaster response scenarios) where the accuracy of solutions is of vital importance.

Bounded Max-Sum is a message-passing algorithm for solving Distributed Constraint Optimization Problems able to compute solutions with a guaranteed approximation ratio. Although its approximate solutions were empirically proved to be within a small percentage of the optimal solution on low and moderate dense problems, in this paper we show that a simple modification systematically provides even better solutions. This is especially relevant in critical applications (e.g. disaster response scenarios) where the accuracy of solutions is of vital importance.

Many real-life optimization problems are combinatorial, i.e. they concern a choice of the best solution from a finite but exponentially
large set of alternatives. Besides, the solution quality of many of these problems can often be evaluated from several points of view
(a.k.a. criteria). In that case, each criterion may be described by a different objective function. Some important and well-known
multicriteria scenarios are:
· In investment optimization one wants to minimize risk and maximize benefits.
· In travel scheduling one wants to minimize time and cost.
· In circuit design one wants to minimize circuit area, energy consumption and maximize speed.
· In knapsack problems one wants to minimize load weight and/or volume and maximize its economical value.
The previous examples illustrate that, in many cases, these multiple criteria are incommensurate (i.e., it is difficult or impossible to
combine them into a single criterion) and conflicting (i.e., solutions that are good with respect one criterion are likely to be bad with
respect to another). Taking into account simultaneously the different criteria is not trivial and several notions of optimality have been
proposed. Independently of the chosen notion of optimality, computing optimal solutions represents an important current research
challenge.
Graphical models are a knowledge representation tool widely used in the Artificial Intelligence field. They seem to be specially
suitable for combinatorial problems. Roughly, graphical models are graphs in which nodes represent variables and the (lack of) arcs
represent conditional independence assumptions. In addition to the graph structure, it is necessary to specify its micro-structure
which tells how particular combinations of instantiations of interdependent variables interact. The graphical model framework
provides a unifying way to model a broad spectrum of systems and a collection of general algorithms to efficiently solve them.
In this Thesis we integrate multi-objective optimization problems into the graphical model paradigm and study how algorithmic
techniques developed in the graphical model context can be extended to multi-objective optimization problems. As we show, multiobjective
optimization problems can be formalized as a particular case of graphical models using the semiring-based framework. It
is, to the best of our knowledge, the first time that graphical models in general, and semiring-based problems in particular are used to
model an optimization problem in which the objective function is partially ordered. Moreover, we show that most of the solving
techniques for mono-objective optimization problems can be naturally extended to the multi-objective context. The result of our work
is the mathematical formalization of multi-objective optimization problems and the development of a set of multiobjective solving
algorithms that have been proved to be efficient in a number of benchmarks.